Normal Nodes and Specht Restriction Filtrations

Definitions and mathematical statement

For a prime $p$, a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ of an integer $n$ is called $p$-regular if no positive part occurs $p$ or more times. Its Young diagram consists of the nodes $(r,c)$ with $1\le r\le\ell$ and $1\le c\le\lambda_r$. A node is removable if deleting it leaves the Young diagram of a partition. These are the usual meanings relevant to the source's phrase "$p$-regular Specht module," which appears to refer to a Specht module labelled by a $p$-regular partition; that interpretation requires confirmation.

In the usual modular setting, $S^\lambda$ denotes a Specht module for the symmetric group $\mathfrak S_n$ over a field of characteristic $p$, and restriction is to $\mathfrak S_{n-1}$. The supplied record does not specify the field or explicitly identify this restriction. More importantly, it does not define the objects $J_i^\lambda$, their construction from the restriction filtration, the ordering of removable nodes, or the precise convention for normal nodes in Kleshchev's sense.

Retaining that unresolved notation, let $A_i$ denote the source's $i$-th removable node. The supplied question asks whether \[ J_i^\lambda\ne J_{i+1}^\lambda \quad\Longleftrightarrow\quad A_i\text{ is normal in Kleshchev's sense}, \] and asks for the minimal $n$ for which this equivalence holds. The admissible indices, endpoint convention, quantification over $p$ and $\lambda$, and meaning of "minimal $n$" are not supplied. In particular, the record does not distinguish a least example from a least rank at which a universal assertion holds or a threshold beyond which it always holds. Consequently, this is a source-faithful statement of the proposed question with unresolved qualifications, not a fully specified conjecture.

Short English statement

The source asks whether jumps in a Specht restriction filtration correspond exactly to normal removable nodes, and asks for a minimal rank $n$. The filtration notation, node conventions, hypotheses, and meaning of the minimum require clarification before a precise quantified statement can be given.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001772$, title "Normal Nodes and Specht Restriction Filtrations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2011.
• Supplied report citation: Mini-Workshop: Modular Representations of Symmetric Groups and Related Objects (2011), Oberwolfach Reports, pp. 1177--1205, DOI 10.4171/owr/2011/21. Report link, also supplied under the literature-source label "original OWR report": https://doi.org/10.4171/owr/2011/21. The report's contents have not been independently inspected for this draft.
• The supplied literature assessment was checked on 2026-08-21. It labels the problem open and reports that the filtration-jump characterization and minimal-$n$ question are posed as open problems, with no complete resolving theorem located. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.